I've asked a similar question a few months ago and explored the OOTB capabilities. It seems that you will have to build it yourself. I see a couple of options here:
Add custom buttons on fallout tasks for an agent to create a related record of the type you want (incident, request etc.), which will be related to the fallout task. You'll have to implement sync between those two records by yourself e.g. status updates, work notes/closure notes, whatever you want to sync. Of course, update the fallout task UI to show related records.
